Transverse classification Transverse homology HFK grid invariant Comparison Effective invariants of transverse knots Lenny Ng Duke University Conference on contact and symplectic topology Universit´ede Nantes 17 June 2011 Partly based on: joint work with Tobias Ekholm, John Etnyre, and Michael Sullivan preliminary joint work with Dylan Thurston These slides available at http://www.math.duke.edu/~ng/nantes.pdf. Transverse classification Transverse homology HFK grid invariant Comparison Outline 1 Transverse classification 2 Transverse homology 3 HFK grid invariant 4 Comparison Transverse classification Transverse homology HFK grid invariant Comparison Transverse knots M cooriented contact 3-manifold with contact structure ξ = ker α. 3 Standard example: M = R , αstd = dz − y dx. Definition A knot K in (M, ξ) is transverse if α> 0 along K (in particular, K ⋔ ξ). Two transverse knots are transversely isotopic if they are isotopic through transverse knots. Transverse classification problem Classify transverse knots of some particular topological type. 3 We’ll restrict our attention to (R , ξstd = ker αstd). Transverse classification Transverse homology HFK grid invariant Comparison Relation to Legendrian knots There is a one-to-one correspondence {transverse knots} ←→{Legendrian knots}/ (+ Legendrian stabilization/destab). In R3, the classical invariant (self-linking number) of T and the classical invariants (Thurston–Bennequin number and rotation number) of L are related by sl(T )= tb(L) − rot(L). Transverse classification Transverse homology HFK grid invariant Comparison Braids and transverse knots Theorem (Bennequin 1983) Any braid (conjugacy class) can be closed in a natural way to 3 produce a transverse knot in (R , ξstd), and every transverse knot is transversely isotopic to a closed braid. Transverse classification Transverse homology HFK grid invariant Comparison Braids and transverse knots Theorem (Bennequin 1983) Any braid (conjugacy class) can be closed in a natural way to 3 produce a transverse knot in (R , ξstd), and every transverse knot is transversely isotopic to a closed braid. Transverse Markov Theorem (Orevkov–Shevchishin 01, Wrinkle 02) Two braids represent the same transverse knot iff related by: conjugation in the braid groups positive stabilization B ←→ Bσn: B B Cf. usual Markov Theorem: topological knots/links are equivalent to braids mod conjugation and positive/negative stabilization. Transverse classification Transverse homology HFK grid invariant Comparison Transverse classification If a transverse knot T is the closure of a braid B, the self-linking number of T is sl(T )= w(B) − n(B) where w(B) = algebraic crossing number of B and n(B) = braid index of B. Definition A topological knot is transversely simple if its transverse representatives are completely determined by self-linking number; otherwise transversely nonsimple. Transverse classification Transverse homology HFK grid invariant Comparison Transverse classification If a transverse knot T is the closure of a braid B, the self-linking number of T is sl(T )= w(B) − n(B) where w(B) = algebraic crossing number of B and n(B) = braid index of B. Definition A topological knot is transversely simple if its transverse representatives are completely determined by self-linking number; otherwise transversely nonsimple. Examples of transversely simple knots: unknot (Eliashberg 1993) torus knots (Etnyre 1999) and the figure 8 knot (Etnyre–Honda 2000) some twist knots (Etnyre–N.–V´ertesi 2010) Transverse classification Transverse homology HFK grid invariant Comparison Transverse nonsimplicity Examples of transversely nonsimple knots: (2, 3)-cable of (2, 3) torus knot (Etnyre–Honda 2003) and other torus knot cables (Etnyre–LaFountain–Tosun 2011) some closed 3-braids (Birman–Menasco 2003, 2008) some twist knots (Etnyre–N.–V´ertesi 2010): number of crossings in shaded region is odd and ≥ 5 Transverse classification Transverse homology HFK grid invariant Comparison Transversely nonsimple knots: Birman–Menasco examples Birman–Menasco 2008: family of knots with braid index 3 that are transversely nonsimple. More precisely, they show that the transverse knots given by the closures of the 3-braids a b c −1 a −1 c b σ1σ2 σ1 σ2 , σ1σ2 σ1 σ2 , which are related by a “negative flype”, are transversely nonisotopic for particular choices of (a, b, c). 5 3 3 −1 5 −1 3 3 σ1σ2σ1σ2 σ1σ2 σ1σ2 Transverse classification Transverse homology HFK grid invariant Comparison Effective transverse invariants Definition A transverse invariant is effective if it can distinguish different transverse knots with the same self-linking number and topological type (i.e., prove that some topological knot is transversely nonsimple). Not known to be effective: Plamenevskaya 2004: distinguished element in Khovanov homology Wu 2005: distinguished elements in Khovanov–Rozansky sln homology N.–Rasmussen 2007: distinguished element in Khovanov–Rozansky HOMFLY-PT homology (known not to be effective) Transverse classification Transverse homology HFK grid invariant Comparison Effective transverse invariants, continued Known to be effective: Ozsv´ath–Szab´o–Thurston 2006: HFK grid invariant: distinguished element in knot Floer homology via grid diagrams Lisca–Ozsv´ath–Stipsicz–Szab´o2008: LOSS invariant: distinguished element in knot Floer homology via open book decompositions Ekholm–Etnyre–N.–Sullivan 2010: transverse homology: filtered version of knot contact homology Transverse classification Transverse homology HFK grid invariant Comparison The conormal construction Idea: use the cotangent bundle to turn smooth topology into symplectic/contact topology. M smooth manifold unit cotangent bundle ST ∗M, which is naturally a contact manifold K ⊂ M embedded submanifold conormal bundle ∗ ∗ N K = {(q, p): q ∈ K, p, v = 0 ∀ v ∈ TqK}⊂ ST M, which is a Legendrian submanifold of ST ∗M. N ∗K ∗ Np K p K Smooth isotopy of K ⊂ M results in Legendrian isotopy of N∗K ⊂ ST ∗M. Transverse classification Transverse homology HFK grid invariant Comparison Knot contact homology ∗ ∗ (K ⊂ M =⇒ N K Legendrian ⊂ ST M contact) Any Legendrian-isotopy invariant of N∗K is a smooth-isotopy invariant of K: for instance, Legendrian contact homology (Eliashberg–Hofer), where defined. For M = Rn, ST ∗M = J1(Sn−1) and LCH is well-defined (Ekholm–Etnyre–Sullivan 05). Definition K ⊂ Rn. The knot contact homology of K is the Legendrian contact homology of N∗K ⊂ ST ∗Rn, ∗ HC∗(K) := LCH∗(N K). In particular, for a knot K ⊂ R3, HC∗(K) is a smooth knot invariant. Transverse classification Transverse homology HFK grid invariant Comparison Form for knot contact homology ∗ ∗ (K ⊂ M =⇒ N K Legendrian ⊂ ST M contact) For a knot K ⊂ R3, the LCH complex for N∗K is a differential graded algebra (CC∗(K),∂) generated by Reeb chords for N∗K, over the group ring ∗ ±1 ±1 R := Z[H1(N K)] =∼ Z[λ , µ ]. The differential counts holomorphic disks in ST ∗R3 with boundary on N∗K. Theorem (N. 2003, 2004, Ekholm–Etnyre–N.–Sullivan in preparation) There is a purely algebraic/combinatorial expression for the DGA (CC∗(K),∂). Transverse classification Transverse homology HFK grid invariant Comparison Holomorphic disks counted in knot contact homology The symplectization R × ST ∗R3, and the Lagrangian cylinder R × N∗K in the symplectization: R × N ∗K R R × ST ∗R3 This holomorphic disk contributes ∂(ai )= aj1 aj2 aj3 + ··· ∗ where ai , aj1 , aj2 , aj3 are Reeb chords of N K. Transverse classification Transverse homology HFK grid invariant Comparison Holomorphic disks counted in knot contact homology The symplectization R × ST ∗R3, and the Lagrangian cylinder R × N∗K in the symplectization: ai R × N ∗K R R × ST ∗R3 aj3 aj1 aj2 This holomorphic disk contributes ∂(ai )= aj1 aj2 aj3 + ··· ∗ where ai , aj1 , aj2 , aj3 are Reeb chords of N K. Transverse classification Transverse homology HFK grid invariant Comparison Properties of knot contact homology (Recall: knot contact homology HC∗(K) = Legendrian contact homology of conormal N∗K ⊂ ST ∗R3.) Theorem (N. 2004) HC∗(K) can be combinatorially shown to be a knot invariant, supported in degrees ∗≥ 0. (Linearized) HC1(K) encodes the Alexander polynomial ∆K (t). HC0(K) detects the unknot. Transverse classification Transverse homology HFK grid invariant Comparison Lifting a contact structure Given a contact manifold (M, ξ), the contact structure ξ itself has a conormal lift to ST ∗M: ∗ ∗ ∗ ∗ N ξ = N+ξ ∪ N−ξ = {(q, p) ∈ ST M : p, v = 0 ∀ v ∈ ξq}. ∗ N+ξ K ∗ STp M p N ∗K ξp ∗ N−ξ If K is transverse to ξ, then the conormal lifts of K and ξ are ∗ ∗ disjoint: N K ∩ N±ξ = ∅. Transverse classification Transverse homology HFK grid invariant Comparison Filtering the LCH differential If K is transverse in (R3, ξ), we can filter the LCH differential for N∗K by counting intersections with the holomorphic 4-manifolds ∗ ∗ 3 R × N±ξ in the symplectization R × ST R . R ∗ × N+ξ R × N ∗K R R ∗ × N−ξ R × ST ∗R3 This lifts the LCH complex from a DGA (A,∂) over R = Z[λ±1, µ±1] to a DGA (A,∂−) over R[U, V ]: e.g. − 2 1 ∂ (ai )= U V aj1 aj2 aj3 + ··· . Transverse classification Transverse homology HFK grid invariant Comparison Filtering the LCH differential If K is transverse in (R3, ξ), we can filter the LCH differential for N∗K by counting intersections with the holomorphic 4-manifolds ∗ ∗ 3 R × N±ξ in the symplectization R × ST R . ai R ∗ × N+ξ R × N ∗K R R ∗ × N−ξ R × ST ∗R3 aj3 aj1 aj2 This lifts the LCH complex from a DGA (A,∂) over R = Z[λ±1, µ±1] to a DGA (A,∂−) over R[U, V ]: e.g. − 2 1 ∂ (ai )= U V aj1 aj2 aj3 + ··· . Transverse classification Transverse homology HFK grid invariant Comparison Transverse homology Definition The (minus) transverse complex of a transverse knot 3 − − K ⊂ (R , ξstd) is the LCH algebra (CT∗ (K)= A,∂ ) over the base ring R[U, V ]= Z[λ±1, µ±1, U, V ], with the differential ∂− ∗ filtered by intersections with N±ξ. This can be viewed as a bi-filtered version of knot contact homology.